Triangular number


A triangular number or triangle number counts objects arranged in an equilateral triangle. The th triangular number is the number of dots in the triangular arrangement with dots on a side, and is equal to the sum of the natural numbers from 1 to. The sequence of triangular numbers, starting at the 0th triangular number, is

Formula

The triangle numbers are given by the following explicit formulas:
where is a binomial coefficient. It represents the number of distinct pairs that can be selected from objects, and it is read aloud as " plus one choose two".
The first equation can be illustrated using a visual proof. For every triangular number, imagine a "half-square" arrangement of objects corresponding to the triangular number, as in the figure below. Copying this arrangement and rotating it to create a rectangular figure doubles the number of objects, producing a rectangle with dimensions, which is also the number of objects in the rectangle. Clearly, the triangular number itself is always exactly half of the number of objects in such a figure, or:. The example follows:
The first equation can also be established using mathematical induction. Since is equal to one, a basis case is established. It follows from the definition that, so assuming the inductive hypothesis for, adding to both sides immediately gives
In other words, since the proposition is true when, and since being true implies that is also true, then the first equation is true for all natural numbers. The above argument can be easily modified to start with, and include, zero.
Carl Friedrich Gauss is said to have found this relationship in his early youth, by multiplying pairs of numbers in the sum by the values of each pair. However, regardless of the truth of this story, Gauss was not the first to discover this formula, and some find it likely that its origin goes back to the Pythagoreans 5th century BC. The two formulae were described by the Irish monk Dicuil in about 816 in his Computus.
The triangular number solves the handshake problem of counting the number of handshakes if each person in a room with people shakes hands once with each person. In other words, the solution to the handshake problem of people is. The function is the additive analog of the factorial function, which is the products of integers from 1 to .
The number of line segments between closest pairs of dots in the triangle can be represented in terms of the number of dots or with a recurrence relation:
In the limit, the ratio between the two numbers, dots and line segments is

Relations to other figurate numbers

Triangular numbers have a wide variety of relations to other figurate numbers.
Most simply, the sum of two consecutive triangular numbers is a square number, with the sum being the square of the difference between the two. Algebraically,
This fact can be demonstrated graphically by positioning the triangles in opposite directions to create a square:
There are infinitely many triangular numbers that are also square numbers; e.g., 1, 36, 1225. Some of them can be generated by a simple recursive formula:
All square triangular numbers are found from the recursion
Also, the square of the th triangular number is the same as the sum of the cubes of the integers 1 to. This can also be expressed as
The sum of the first triangular numbers is the th tetrahedral number:
More generally, the difference between the th -gonal number and the th -gonal number is the th triangular number. For example, the sixth heptagonal number minus the sixth hexagonal number equals the fifth triangular number, 15. Every other triangular number is a hexagonal number. Knowing the triangular numbers, one can reckon any centered polygonal number; the th centered -gonal number is obtained by the formula
where is a triangular number.
The positive difference of two triangular numbers is a trapezoidal number.

Other properties

Triangular numbers correspond to the first-degree case of Faulhaber's formula.
Alternating triangular numbers are also hexagonal numbers.
Every even perfect number is triangular, given by the formula
where is a Mersenne prime. No odd perfect numbers are known; hence, all known perfect numbers are triangular.
For example, the third triangular number is 6, the seventh is 28, the 31st is 496, and the 127th is 8128.
In base 10, the digital root of a nonzero triangular number is always 1, 3, 6, or 9. Hence, every triangular number is either divisible by three or has a remainder of 1 when divided by 9:
The digital root pattern for triangular numbers, repeating every nine terms, as shown above, is "1, 3, 6, 1, 6, 3, 1, 9, 9".
The converse of the statement above is, however, not always true. For example, the digital root of 12, which is not a triangular number, is 3 and divisible by three.
If is a triangular number, then is also a triangular number, given is an odd square and
will always be a triangular number, because, which yields all the odd squares are revealed by multiplying a triangular number by 8 and adding 1, and the process for given is an odd square is the inverse of this operation.
The first several pairs of this form are:,,,,,, … etc. Given is equal to, these formulas yield,,,, and so on.
The sum of the reciprocals of all the nonzero triangular numbers is
This can be shown by using the basic sum of a telescoping series:
Two other formulas regarding triangular numbers are
and
both of which can easily be established either by looking at dot patterns or with some simple algebra.
In 1796, German mathematician and scientist Carl Friedrich Gauss discovered that every positive integer is representable as a sum of three triangular numbers, writing in his diary his famous words, "ΕΥΡΗΚΑ! ". This theorem does not imply that the triangular numbers are different, nor that a solution with exactly three nonzero triangular numbers must exist. This is a special case of the Fermat polygonal number theorem.
The largest triangular number of the form is 4095.
Wacław Franciszek Sierpiński posed the question as to the existence of four distinct triangular numbers in geometric progression. It was conjectured by Polish mathematician Kazimierz Szymiczek to be impossible and was later proven by Fang and Chen in 2007.
Formulas involving expressing an integer as the sum of triangular numbers are connected to theta functions, in particular the Ramanujan theta function.

Applications

A fully connected network of computing devices requires the presence of cables or other connections; this is equivalent to the handshake problem mentioned above.
In a tournament format that uses a round-robin group stage, the number of matches that need to be played between teams is equal to the triangular number. For example, a group stage with 4 teams requires 6 matches, and a group stage with 8 teams requires 28 matches. This is also equivalent to the handshake problem and fully connected network problems.
One way of calculating the depreciation of an asset is the sum-of-years' digits method, which involves finding, where is the length in years of the asset's useful life. Each year, the item loses, where is the item's beginning value, is its final salvage value, is the total number of years the item is usable, and the current year in the depreciation schedule. Under this method, an item with a usable life of = 4 years would lose of its "losable" value in the first year, in the second, in the third, and in the fourth, accumulating a total depreciation of of the losable value.

Triangular roots and tests for triangular numbers

By analogy with the square root of, one can define the triangular root of as the number such that :
which follows immediately from the quadratic formula. So an integer is triangular if and only if is a square. Equivalently, if the positive triangular root of is an integer, then is the th triangular number.